The accurate assessment of a Wireless Power Transfer (WPT) system's magnetic field distribution and the consequent mutual inductance (M) is essential to its dependable and effective functioning. A thorough derivation of these basic parameters is presented in this paper, beginning with the time-harmonic Maxwell’s equations under the quasi-static approximation. The analysis yields the magnetic vector potential (A) controlling Partial Differential Equation (PDE), which reduces to the vector version of the Poisson Equation: ∇ 2 A = −μJ. It shows that the current density J, scaled by the magnetic permeability μ of the medium, directly drives the spatial curvature of the magnetic vector potential. This PDE directly connects coil shape, material characteristics, and current excitation to the resultant magnetic field distribution that controls mutual inductance and WPT efficiency. It is the fundamental equation solved in numerical electromagnetic tools like FEM solvers. The flux linkage (φ) is established using the formal solution to this PDE, leading to the famous Neumann Formula for mutual inductance M, the essential analytical descriptor of coil coupling. In addition, the work addresses the need to solve the PDE for complicated coil geometries using numerical methods like the Finite Element Method (FEM), with a focus on the Double-D Quadrupole (DDQ) structure designed for improved misalignment tolerance. The last metric for structural design validation is the resultant coupling coefficient (k), which establishes a clear connection between electromagnetic theory and WPT system performance.
Amroliya et al. (Fri,) studied this question.